Edwin Encarnacion hit two doubles and scored a run for the Blue Jays (55-65), while starter Esmil Rogers (3-7) pitched six innings with six strikeouts, five hits and an earned run. Reliever Brett Cecil ruined Rogers' chance at a win though, blowing Toronto's 3-1 lead in the ninth inning.
Brad Lincoln pitched an inning and a third in relief to earn the win for Toronto.
Designated hitter David Ortiz hit a home run for Boston (72-50). Jon Lester (10-7) gave up three runs — two earned — and fanned three hitters over six and a third innings of work. Brandon Workman (3-2) pitched three innings for the Red Sox and took the loss.
Davis drew cheers when he led off the bottom of the 10th with his third double of the game. He then advanced to third base on Jose Bautista's groundout to short.
Workman then intentionally walked Encarnacion to put runners on the corners. With Brett Lawrie at the plate, Encarnacion stole second base.
Lawrie then hit a sharp single off the glove of Boston shortstop Stephen Drew to drive home Davis and end the game.
Davis opened the scoring in the third inning when he hit a sharp single to Lester, where the pitcher fumbled the ball. Lester's throw to first went past Mike Napoli, allowing the speedy Toronto outfielder to round second. Boston right-fielder Shane Victorino misthrew to third baseman Will Middlebrooks, allowing Davis to sprint home for a 1-0 lead.
Blue Jays' right-fielder Jose Bautista ended the inning by grounding out to third.
After a double play gave Boston two outs in the top of the fourth, second baseman Munenori Kawasaki stabbed Napoli's sharp hit out of the air to end the inning.
The Blue Jays added to their lead in the bottom of the fourth. Encarnacion hit a double up the third-base line and was driven in when designated hitter Mark DeRosa hit a double to shallow right field. Napoli looked foolish on DeRosa's hit, backing up to the outfield from first base and jumping for the ball, but missing the play entirely.
DeRosa was stranded on second when J.P. Arencibia flied out to left and left-fielder Kevin Pillar — making his major league debut — lined out to right.
Both teams stranded runners in scoring position in the fifth inning, maintaining the Blue Jays' 2-0 lead.
Ortiz pulled Boston to within one on a home run to deep right field on a 3-2 count versus Rogers in the sixth inning. Pillar made an impressive diving catch in left field for his first put out in the majors. Rogers then caught a comebacker by Jonny Gomes to end the inning.
Encarnacion hit a second double to lead off the bottom of the sixth, and Lawrie reached first after being hit by a pitch. Encarnacion was moved to third when DeRosa grounded into a double play.
Arencibia made it 3-1 for Toronto with a double off the left-field wall, scoring Encarnacion. The sixth inning ended when Pillar grounded out.
Reliever Neil Wagner took the mound for the Blue Jays in the seventh, quickly retiring three Red Sox hitters.
Kawasaki reached first base on another error by Lester when the Toronto infielder's hit bounced to the mound and the pitcher misplayed it. Lester recovered by striking out Reyes, but Red Sox manager John Farrell still pulled the starting pitcher in favour of right-hander Brandon Workman.
The Boston reliever struck out Davis, then induced a pop fly to left field by Bautista.
Wagner continued his efficient pitching in the eighth, retiring Ellsbury, Victorino and Dustin Pedroia in quick succession. The right-handed pitcher threw two scoreless innings with three strikeouts.
Workman was also effective for the Red Sox, striking out Encarnacion and getting Lawrie to fly out to right field. Adam Lind pinch hit for DeRosa, drawing a walk. The inning ended when Arencibia struck out swinging.
Cecil took the mound in the ninth and his first out was gifted to him when Ortiz — not known for his speed— tried to beat out a defensive shift by bunting. The lumbering DH was thrown out at first by Arencibia.
Gomes reached base when he was hit by a pitch, but Cecil settled down to fan Stephen Drew.
Napoli made Cecil pay for plunking Gomes, driving the ball over the right-field fence and tying the game 3-3 with two out in the ninth. It was Napoli's first home run since July 24.
The reliever quickly unravelled, giving up a double to Jarrod Saltalamacchia, with Middlebrooks reaching first after being hit by a pitch. Cecil was pulled in favour of Brad Lincoln after giving up a single to Ellsbury.
Lincoln got the Blue Jays out of the inning when Victorino flied out to left field.
Workman forced extra innings by dispatching three Toronto hitters in a row.
Lincoln stayed on the mound for the 10th, with Pedroia reaching base on Lawrie's fielding error to start the inning. After Ortiz flew out to left field, Lincoln fanned Gomes and got Napoli to ground out to third, setting the stage for Lawrie's game-winning hit.
Notes: Kawasaki and Pillar were called up from triple-A Buffalo to take the places of utility player Emilio Bonifacio and centre-fielder Colby Rasmus on the roster. Bonifacio was traded to Kansas City earlier in the day for a player to be named later or cash, while Rasmus was placed on the 15-day disabled list.
Follow John Chidley-Hill on Twitter @jchidleyhillSuggest a correction